Bouncing a point between two simple circles automatically creates infinite, complex fractals out of thin air.
April 16, 2026
Original Paper
From Tangency to Fractals: Quadratic Dynamics in Nested Convex Geometry
arXiv · 2604.09872
The Takeaway
We usually think of fractals as complex computer-generated art or rare occurrences in nature, but this research shows they are built into the very DNA of basic geometry. By simply projecting points between two nested shapes—like an oval inside a circle—and following their curves, the math 'contracts' into a self-similar fractal pattern. You don't need a supercomputer or a complex recursive formula; you just need two shapes and a simple rule for how they touch. This means that the infinite complexity we see in nature might not be a result of complex processes, but a natural byproduct of simple shapes interacting. It suggests that if you zoom in far enough on almost any curved boundary in the universe, you're likely to find a fractal hiding there.
From the abstract
We study the dynamics generated by return maps associated with nested convexbodies and growing domains satisfying the geometric normal property in the plane.These maps are defined by transporting boundary points along normal directions tothe surrounding domain and projecting them back onto the boundary of a subsequentconvex set.We introduce a tangency condition between consecutive convex sets and showthat it cancels the linear term in the local expansion of the transition operators. Asa result,